Like in the original games, after you get all 16 badges from Johto & Kanto, you can go to Professor Oak. He tells you that Mt. Silver is now accessible for you and gives you HM08 - Rock Climb. Go there through Route 22 and you can travel up to Mt. Silver. If, when you get into the cave, you take the east path, climbing up the cave with Rock Climb, you will eventually get to the peak of Mt. Silver where Red is stationed.

Red is the highest levelled trainer of any Pokémon game so you have to be prepared for an epic fight. Luckily, as you are on top of a mountain, the Hail Weather Effect is running so that can help you in taking out 5 of his Pokémon as his Lapras is immune. Once defeated, the game cuts to the credits and further things are unlocked in the game.